346 XI ELEMENTARY SPECTRAL THEORY operator, which, as is readily verified, corresponds to the kernel function KO, 0 - £ Aft 9h(s)(ph(t}, where h runs through all the indices (in finite num- h ber) such that \ < 0. The conclusion is then immediate. (11.6.9) We can consider the operator U in a larger prehilbert space, namely the space F+ of regulated functions (Section 7.6) which are continuous on the right (i.e. such (hat/(f +) =/(/) for a < t < b) and such that f(b) = 0; for such a function the relation | \f(t)\2 dt = Q implies f(t) = 0 everywhere Ja in I = [a, b], for it implies f(t) = 0 except at the points of a denumerable subset D (by (8.5.3)), and every t such that a < t < b is limit of a decreasing sequence of points of I — D. The space G may be identified to a subspace of F+ , by changing eventually the value of a continuous function /e G at the point 6; it is easily proved (using (7.6.1)) that G is dense in F+ . The argument of (11.2.8) then shows that U is a compact mapping o/F+ into the Banach space E = ^C(I) (and a fortiori a compact mapping of the prehilbert space F+ into itself). All the results proved for the operator U in G are still valid (with their proofs) when G is replaced by F+ . PROBLEMS 1. Extend the results of Section 11 .6 (with the exception of (11 .6.7)) to the case in which K(s, t) satisfies the assumptions of Section 8.11, Problem 4 (use that problem, as well as Section 11.2, Problem 5). 2. In the prehilbert space G of Section 11.6, let (/„) be a total orthonormal system (Section 6.5); let K.fo t) = Z/*(*WO and Hn(s) - f |Kn(*, t)\ dt k=l Ja (the ">zth Lebesgue function" of the orthonormal system (/„)). For any function g e G, let $n(g) = £ (0|/*)A, so that sn(g)(x) - P »!„(*, t)g(t) dt for any x e I. *=1 Ja (a) Prove that if, for an x0 e I, the sequence (Hn(x0)) is unbounded, then there exists a function g e G such that the sequence (sn(g)(xQ)) is unbounded. (Use contradiction, and show that under the contrary assumption it is possible to define a strictly increasing sequence of integers (nk), and a sequence (gk) of functions of G, with the follow- ch = sup I J n |Jfl ing properties: (1) let ch = sup Kn(x01 t)gh(t)dt (a number which is finite by rb assumption), let dk — ci + c2 H- • • • + ck-i, let mk = |Knfc(^0 , t) dt, and let qk = sup(mi, . . ., wfc_i); then mk ^ 2k+1(gk -f l)(dk + k); (2) let